# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a242640 Showing 1-1 of 1 %I A242640 #9 May 21 2014 18:04:54 %S A242640 1,3,5,4,7,10,7,13,17,21,6,11,16,21,26,12,23,32,38,44,50,8,15,22,29, %T A242640 36,43,50,15,29,41,53,61,69,77,85,13,25,37,46,55,64,73,82,91,18,35,50, %U A242640 65,80,90,100,110,120,130,12,23,34,45,56,67,78,89,100,111,122,28,55,80,102,120,138,150,162,174,186,198,210 %N A242640 Triangle read by rows: T(s,n) (1 <= s <= n) = Sum_{d|n, d <= s} d^2 + s*Sum_{d|n, d>s} d. %D A242640 P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table I. Note that the entry 53 should be 50. %e A242640 Triangle begins: %e A242640 [1] %e A242640 [3, 5] %e A242640 [4, 7, 10] %e A242640 [7, 13, 17, 21] %e A242640 [6, 11, 16, 21, 26] %e A242640 [12, 23, 32, 38, 44, 50] %e A242640 [8, 15, 22, 29, 36, 43, 50] %e A242640 [15, 29, 41, 53, 61, 69, 77, 85] %e A242640 ... %e A242640 The full array (see A242639) begins: %e A242640 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ... %e A242640 1, 5, 7, 13, 11, 23, 15, 29, 25, 35, 23, 55, ... %e A242640 1, 5, 10, 17, 16, 32, 22, 41, 37, 50, 34, 80, ... %e A242640 1, 5, 10, 21, 21, 38, 29, 53, 46, 65, 45, 102, ... %e A242640 1, 5, 10, 21, 26, 44, 36, 61, 55, 80, 56, 120, ... %e A242640 1, 5, 10, 21, 26, 50, 43, 69, 64, 90, 67, 138, ... %e A242640 1, 5, 10, 21, 26, 50, 50, 77, 73, 100, 78, 150, ... %e A242640 1, 5, 10, 21, 26, 50, 50, 85, 82, 110, 89, 162, ... %e A242640 ... %p A242640 with(numtheory): %p A242640 A:=proc(s,n) local d,s1,s2; %p A242640 s1:=0; s2:=0; %p A242640 for d in divisors(n) do %p A242640 if d <= s then s1:=s1+d^2 else s2:=s2+d; fi; od: %p A242640 s1+s*s2; end; %p A242640 for n from 1 to 15 do lprint([seq(A(s,n),s=1..n)]); od: %Y A242640 Upper triangle of array in A242639. %K A242640 nonn,tabl %O A242640 1,2 %A A242640 _N. J. A. Sloane_, May 21 2014 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE